The Experts below are selected from a list of 888 Experts worldwide ranked by ideXlab platform
Lars Chittka - One of the best experts on this subject based on the ideXlab platform.
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Radar Tracking and Motion-Sensitive Cameras on Flowers Reveal the Development of Pollinator Multi-Destination Routes over Large Spatial Scales
PLoS Biology, 2012Co-Authors: Mathieu Lihoreau, Andrew M Reynolds, Nigel E. Raine, Ralph Stelzer, Ka Lim, Allan Smith, Juliet L. Osborne, Lars ChittkaAbstract:Central place foragers, such as pollinating bees, typically develop circuits (traplines) to visit multiple foraging sites in a manner that minimizes overall travel distance. Despite being taxonomically widespread, these routing behaviours remain poorly understood due to the difficulty of tracking the foraging history of animals in the wild. Here we examine how bumblebees (Bombus terrestris) develop and optimise traplines over large spatial scales by setting up an array of five artificial flowers arranged in a Regular Pentagon (50 m side length) and fitted with motion-sensitive video cameras to determine the sequence of visitation. Stable traplines that linked together all the flowers in an optimal sequence were typically established after a bee made 26 foraging bouts, during which time only about 20 of the 120 possible routes were tried. Radar tracking of selected flights revealed a dramatic decrease by 80% (ca. 1500 m) of the total travel distance between the first and the last foraging bout. When a flower was removed and replaced by a more distant one, bees engaged in localised search flights, a strategy that can facilitate the discovery of a new flower and its integration into a novel optimal trapline. Based on these observations, we developed and tested an iterative improvement heuristic to capture how bees could learn and refine their routes each time a shorter route is found. Our findings suggest that complex dynamic routing problems can be solved by small-brained animals using simple learning heuristics, without the need for a cognitive map.
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Radar Tracking and Motion-Sensitive Cameras on Flowers Reveal the Development of Pollinator Multi- Destination Routes over Large Spatial Scales
2012Co-Authors: Nigel E. Raine¤b, Andrew M Reynolds, Ralph J. Stelzer, Ka S. Lim, Alan D. Smith, Juliet L. Osborne¤c, Lars ChittkaAbstract:Central place foragers, such as pollinating bees, typically develop circuits (traplines) to visit multiple foraging sites in a manner that minimizes overall travel distance. Despite being taxonomically widespread, these routing behaviours remain poorly understood due to the difficulty of tracking the foraging history of animals in the wild. Here we examine how bumblebees (Bombus terrestris) develop and optimise traplines over large spatial scales by setting up an array of five artificial flowers arranged in a Regular Pentagon (50 m side length) and fitted with motion-sensitive video cameras to determine the sequence of visitation. Stable traplines that linked together all the flowers in an optimal sequence were typically established after a bee made 26 foraging bouts, during which time only about 20 of the 120 possible routes were tried. Radar tracking of selected flights revealed a dramatic decrease by 80 % (ca. 1500 m) of the total travel distance between the first and the last foraging bout. When a flower was removed and replaced by a more distant one, bees engaged in localised search flights, a strategy that can facilitate the discovery of a new flower and its integration into a novel optimal trapline. Based on these observations, we developed and tested an iterative improvement heuristic to capture how bees could learn and refine their routes each time a shorter route is found. Our findings suggest that complex dynamic routing problems can be solved by small-brained animals using simple learning heuristics, without the need for
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rinciple of the iterative improvement heuristic for flight path optimization.
2012Co-Authors: Mathieu Lihoreau, Andrew M Reynolds, Nigel E. Raine, Juliet L. Osborne, Ralph J. Stelzer, Ka S. Lim, Alan D. Smith, Lars ChittkaAbstract:At each stage, a model bee chooses to move between flowers according to six assumptions: (1) the bee can uniquely identify each flower; (2) the bee has a finite probability of using transition vectors joining each pair of flowers; (3) the initial probability of using a vector depends on the distance between the two flowers (in our simulations nearest neighbour flowers are visited with a probability = 0.6 and more distant locations are visited with a probability = 0.1); (4) the bee computes the net length of the route travelled by summing the distances of all vectors comprising the flower visit sequence; (5) having completed a route passing through all the flowers at least once, the bee compares the net length of the current route to the net length of the shortest route experienced so far; (6) if the new route is shorter, the probabilities of using the vectors forming this new route in the next foraging bout are increased by a common factor (in our simulation, a factor of 2). The figure illustrates a late stage of trapline development between five flowers arranged in a Regular Pentagon (where three different paths starting and finishing at the nest-box have been selected over time (N12354N, N13245N, N12345N)). Strengthening the vectors forming the shortest route (N12345N) makes this route more “attractive” (the thickness of the arrow is proportional to the probability of the vector being used). As the bee is more likely to take the shortest route, longer routes will be gradually abandoned (see simulations in Figure S3).
Serge Tabachnikov - One of the best experts on this subject based on the ideXlab platform.
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Periodic trajectories in the Regular Pentagon
2016Co-Authors: Dmitry Fuchs, Serge TabachnikovAbstract:In our recent paper [1], we studied periodic billiard trajectories in a Regular Pentagon and in the isosceles triangle with with the angles (pi/5, pi/5, 3pi/5). We provided a full computation of the lengths of these trajectories, both geo-metric and combinatoric, and formulated some conjectures concerning symboli
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Periodic trajectories in the Regular Pentagon, II
Moscow Mathematical Journal, 2013Co-Authors: Fuchs Dmitry, Serge TabachnikovAbstract:We study periodic linear trajectories in the double Pentagon and periodic billiard trajectories in the Regular Pentagon.
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Periodic Trajectories in the Regular Pentagon, II
Moscow Mathematical Journal, 2013Co-Authors: Dmitry Fuchs, Serge TabachnikovAbstract:In our recent paper [1], we studied periodic billiard trajectories in a Regular Pentagon and in the isosceles triangle with with the angles (π/5, π/5, 3π/5). We provided a full computation of the lengths of these trajectories, both geometric and combinatoric, and formulated some conjectures concerning symbolic periodic trajectories. The main goal of this article is to prove two of these conjectures. Technically, the study of billiard trajectories in a Regular Pentagon is essentially equivalent to the study of geodesics in the “double Pentagon,” a translation surface obtained from two centrally symmetric copies of a Regular Pentagon by pairwise pasting the parallel sides. The result is a surface of genus 2 that has a flat structure inherited from the plane and a conical singularity. See [2, 3, 4, 5, 6, 7] for surveys of flat surfaces and rational polygonal billiards. Let us describe the relevant results from [1]. First of all, a periodic linear trajectory is always included into a parallel family of such trajectories, and when we talk about the period, length, symbolic orbit, etc., we always mean these parallel families. See Figure 1. Second, the double Pentagon has an involution, the central symmetry that exchanges the two copies of the Regular Pentagon. This involution interchanges the linear trajectories that have the opposite directions. For this reason, we identify the opposite directions, so the set of directions is the real projective line RP. We identify this projective line with the circle at infinity of the hyperbolic plane in the Poincare disc model. It is clear from Figure 1 that the directions of the sides of the Pentagons are periodic: every linear trajectory in this direction is closed. In fact, the socalled Veech dichotomy applies to the double Pentagon: if there exists a periodic trajectory in some direction then all parallel trajectories are also periodic (and they form two strips, longer – shaded in Figure 1, and shorter – left unshaded,
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Periodic trajectories in the Regular Pentagon, II
arXiv: Dynamical Systems, 2011Co-Authors: Dmitry Fuchs, Serge TabachnikovAbstract:In our recent paper, we studied periodic billiard trajectories in the Regular Pentagon and closed geodesic on the double Pentagon, a translation surface of genus two. In particular, we made a number of conjectures concerning symbolic periodic trajectories. In this paper, we prove two of these conjectures.
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Periodic trajectories in the Regular Pentagon, II
2011Co-Authors: Dmitry Fuchs, Serge TabachnikovAbstract:1 Introduction and formulation of results In our recent paper [1], we studied periodic billiard trajectories in a Regular Pentagon and in the isosceles triangle with with the angles (π/5, π/5, 3π/5). We provided a full computation of the lengths of these trajectories, both geometric and combinatoric, and formulated some conjectures concerning symboli
Hrachya S Harutyunyan - One of the best experts on this subject based on the ideXlab platform.
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the chord length distribution function for Regular polygons
Advances in Applied Probability, 2009Co-Authors: Hrachya S Harutyunyan, V. K. OhanyanAbstract:In this paper we obtain an elementary expression for the chord length distribution function of a Regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a Regular triangle, a square, a Regular Pentagon, and a Regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.
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chord length distribution of Pentagonal and hexagonal rods relation to small angle scattering
Journal of Applied Crystallography, 2009Co-Authors: Wilfried Gille, Narine G Aharonyan, Hrachya S HarutyunyanAbstract:Based on explicit formulas of chord length density functions (CLDs) for a Regular Pentagon and a hexagon, the CLDs of infinitely long Regular homogeneous Pentagonal/hexagonal cylinders are discussed. Characteristic properties of the small-angle scattering of these cylinders are studied.
Mauro Maria Baldi - One of the best experts on this subject based on the ideXlab platform.
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japanese temple geometry a digital sangaku about a Regular Pentagon and the golden ratio
Social Science Research Network, 2016Co-Authors: Amelia Carolina Sparavigna, Mauro Maria BaldiAbstract:Sangaku are wooden tablets depicting geometric or mathematical problems. They are objects typical of the Edo period. Since these tables were exposed in the temples, the related geometry is known as the Japanese Temple Geometry. Here we illustrate a digital approach to Sangaku. The specific problem we are discussing in the construction of a Regular Pentagon.
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A Study of the Regular Pentagon with a Classic Geometric Approach
2016Co-Authors: Amelia Carolina Sparavigna, Mauro Maria BaldiAbstract:In this paper we will consider a Regular Pentagon and discuss three of its properties, which are linking side, radius, diagonal and apothem to the golden ratio. One of the properties, that regarding the ratio between the diagonal and the radius of the circumscribed circumference is strictly connected to the construction of the Regular Pentagon with compass and straightedge.
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Symmetry and Golden Ratio in the Analysis of Regular Pentagon
2016Co-Authors: Amelia Carolina Sparavigna, Mauro Maria BaldiAbstract:Regular Pentagon is a geometric figure which had a relevant symbolic meaning in Pythagorean and Platonic philosophies and a subsequent important role in the Western thought, appearing also in arts and architecture. A property of the Pentagon, which was probably found by Pythagoreans, is that the ratio between diagonal and side is the golden ratio. Here, we will study some links between Regular Pentagon and golden ratio. We will focus first on the group of five-fold rotation symmetry, to find the position of the vertices of this geometric figure in the complex plane; then, we will give an analytic method to solve the same problem in the Cartesian plane. In the analytic approach, we find the golden ratio without any specific geometric consideration. This study can be interesting also for a general education in mathematics, because it can convey and link several concepts, requiring only a general pre-college education.
Mathieu Lihoreau - One of the best experts on this subject based on the ideXlab platform.
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Radar Tracking and Motion-Sensitive Cameras on Flowers Reveal the Development of Pollinator Multi-Destination Routes over Large Spatial Scales
PLoS Biology, 2012Co-Authors: Mathieu Lihoreau, Andrew M Reynolds, Nigel E. Raine, Ralph Stelzer, Ka Lim, Allan Smith, Juliet L. Osborne, Lars ChittkaAbstract:Central place foragers, such as pollinating bees, typically develop circuits (traplines) to visit multiple foraging sites in a manner that minimizes overall travel distance. Despite being taxonomically widespread, these routing behaviours remain poorly understood due to the difficulty of tracking the foraging history of animals in the wild. Here we examine how bumblebees (Bombus terrestris) develop and optimise traplines over large spatial scales by setting up an array of five artificial flowers arranged in a Regular Pentagon (50 m side length) and fitted with motion-sensitive video cameras to determine the sequence of visitation. Stable traplines that linked together all the flowers in an optimal sequence were typically established after a bee made 26 foraging bouts, during which time only about 20 of the 120 possible routes were tried. Radar tracking of selected flights revealed a dramatic decrease by 80% (ca. 1500 m) of the total travel distance between the first and the last foraging bout. When a flower was removed and replaced by a more distant one, bees engaged in localised search flights, a strategy that can facilitate the discovery of a new flower and its integration into a novel optimal trapline. Based on these observations, we developed and tested an iterative improvement heuristic to capture how bees could learn and refine their routes each time a shorter route is found. Our findings suggest that complex dynamic routing problems can be solved by small-brained animals using simple learning heuristics, without the need for a cognitive map.
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rinciple of the iterative improvement heuristic for flight path optimization.
2012Co-Authors: Mathieu Lihoreau, Andrew M Reynolds, Nigel E. Raine, Juliet L. Osborne, Ralph J. Stelzer, Ka S. Lim, Alan D. Smith, Lars ChittkaAbstract:At each stage, a model bee chooses to move between flowers according to six assumptions: (1) the bee can uniquely identify each flower; (2) the bee has a finite probability of using transition vectors joining each pair of flowers; (3) the initial probability of using a vector depends on the distance between the two flowers (in our simulations nearest neighbour flowers are visited with a probability = 0.6 and more distant locations are visited with a probability = 0.1); (4) the bee computes the net length of the route travelled by summing the distances of all vectors comprising the flower visit sequence; (5) having completed a route passing through all the flowers at least once, the bee compares the net length of the current route to the net length of the shortest route experienced so far; (6) if the new route is shorter, the probabilities of using the vectors forming this new route in the next foraging bout are increased by a common factor (in our simulation, a factor of 2). The figure illustrates a late stage of trapline development between five flowers arranged in a Regular Pentagon (where three different paths starting and finishing at the nest-box have been selected over time (N12354N, N13245N, N12345N)). Strengthening the vectors forming the shortest route (N12345N) makes this route more “attractive” (the thickness of the arrow is proportional to the probability of the vector being used). As the bee is more likely to take the shortest route, longer routes will be gradually abandoned (see simulations in Figure S3).